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## Homework Statement

The right angle boom which supports the 230-kg cylinder is supported by three cables and a ball-and-socket joint at

*O*attached to the vertical

*x-y*surface. Determine the reactions at

*O*and the cable tensions.

## Homework Equations

M [/B]=

**r**x

**F**

unit vector = (vector)/(magnitude of vector)

weight = weight of hanging mass = 230*9.81 = 2256.3 N

## The Attempt at a Solution

So far, I've gotten two of the reaction forces and two of the cable tensions. Using unit vectors, I was able to solve for the tension vectors:

**T_ac**= <-.4745, .4745, -.7414>T_ac

**T_bd**= <0, .7880, -.6156>T_bd

**T_be**= <0, 0, -1>T_be

Because the moment about an axis sums to zero:

Σ

**M_x**=

**0**

0 = (weight of mass - .4745*T_ac - .7880*T_bd)*radius from axis

T_ac = (1/.4745)(2256.3 - .7880*T_bd)

Σ

**M_z**=

**0**

0 = (T_bd*.7880 - .5*weight)*radius from axis

T_bd = 1431.66 N which goes to 1430 N (The homework site I'm using only allows three significant digits.)

T_ac = (1/.4745)(2256.3 - .7880*T_bd)

T_ac = 2377.56 N which goes to 2380 N

Up until this point, all the forces were equidistant from the axis in question.

Somewhere in here my numbers get messed up - the previous two tension forces are correct, but this one is wrong.

**Σ**

**M_y**=**0**0 = T_ac*-.4745*2.5 - T_be*-1*1.9 - T_be*1.9 + .6156*1.9*T_bd

T_be = (1/1.9)(T_ac*.4745*2.5 - .6156*1.9*T_bd)

T_be = 606 N

Solving for the reaction forces:

ΣF_x = 0

0 = T_ac*-.4745 + O_x

O_x = 1130 N

ΣF_y = 0

0 = O_y + .4745*T_ac + T_bd*.7880 - weight

O_y = 0 N

These two reaction forces are correct, so if I'm doing something wrong up to this point, I'm making compensating errors. The next reaction force is incorrect, but it depends on T_be and since I know that number is wrong, even if my method here is correct, I won't be able to get the right answer.

ΣF_z = 0

0 = -.7414*T_ac - .6156*T_bd - T_be + O_z

O_z = 3250 N

I'm not sure what I'm doing wrong, but at this point I'd be happy to find any mistake!

Thanks for your time!